Pivoted shoe thrust bearings have long been used in high speed applications or where low friction losses and low wear rates are essential. An example of such a bearing is the Kingsbury thrust bearing, or Michell bearing in Europe, where the bearing members are pivotable shoes which rest on hard steel pivots in a bearing housing. The shoes are free to automatically form a wedge-shaped oil film between the shoe surface and the moving thrust collar. The thrust collar transmits the thrust force through the hydrodynamic oil film to the pivoted shoes. In the prior art bearing, a base ring supports the shoes and equalizes the shoe loading. A housing is provided to contain and support the internal bearing elements. A shoe cage restrains the shoes against movement with the thrust collar, but not against outward displacement. The thrust load of the bearing is finally transmitted to a machine frame connected to the housing.
The conventional pivoted shoe thrust bearing also includes a lubricating system which continuously supplies the thrust collar and shoes with lubricating oil. In some applications, a cooling system is provided to reduce the temperature rise in the bearing.
Prior art pivoted shoes have had a flat surface on one side and a pivoting mechanism on the other side. One such pivoting mechanism in the prior art is the convex surface with an offset center of radius of curvature which provides line contact with a supporting surface. Another prior art method of pivoting is the point contact system, where the shoe has a hardened insert in the back which allows the shoe to pivot slightly. If the location of the pivot coincides with the geometrical center of the shoe, it becomes a centrally pivoted thrust bearing. Centrally pivoted bearings are useful in marine and other applications where reversibility is required.
The pivoted shoe in its pivoted position creates a tapered oil film between the shoe and the thrust collar. The oil film provides hydrodynamic pressure and load carrying capacity. The maximum load carrying capacity of the bearing is dependent on the inclination of the shoe and the location of the pivot point on the back of the shoe. The inclination of the shoe is usually designated by the symbol .alpha., which represents the ratio of the maximum film thickness h.sub.1 to a minimum film thickness h.sub.0.
Earlier investigations on fixed, as opposed to pivoted, shoe bearings [References 1-4] provided optimum values of .alpha. required for maximum load carrying capacity for a few oil film shapes such as taper, step, exponential and polynomial by solving a one dimensional Reynolds equation. The conclusions drawn on the basis of one dimensional analysis [Reference 3] underestimated the importance of film shape effect on the performance characteristics of fixed shoe bearings. The design variable .alpha. for a fixed shoe bearing is not of much practical value for a practicing engineer in industry as it is a function of minimum film thickness, which is controlled by the load. Therefore, fixed shoe bearings cannot be designed for these optimum values of .alpha., unless the load is strictly fixed, and this limits the applications of such a design.
To overcome this problem, the pivoted shoe bearing, with a flat surface, became the subject of basic developments in hydrodynamic lubrication of bearings. In the case of pivoted shoe bearings, the maximum to minimum film thickness ratio .alpha., is controlled by the location of pivot position and is independent of the minimum film thickness. This feature of a pivoted shoe bearing is the basic cause of its increasing popularity in the field of thrust and even journal bearings.
One dimensional flow solutions were modified by using correction factors to account for the effect of side leakage in finite bearings [References 5-7]. These correction factors were determined experimentally [References 5, 6]. Computer-aided finite difference solutions of a two dimensional Reynolds equation provided performance charts to analyze pivoted shoe bearings with flat surfaces [References 8-11]. The load carrying capacity of pivoted shoe thrust bearings has also been studied with reference to a few convex surface profiles [References 12-14].
Optimum solutions to many physical problems, such as the optimum path of a particle in motion, the optimum profile for sound traveling through horns [Reference 15], and the optimum shape profile for concentrators used in ultrasonic machining [Reference 16], have yielded cycloidal and catenoidal surface profiles. Although excellent performance based on one dimensional flow analysis of discontinuous oil film shape caused by a step profile provided enough incentive to researchers for the extension of related research in the past [Referrences 17-21], the present invention is directed towards one dimensional continuous surface profiles.
Exact solutions are known to the Reynolds equation for two dimensional flow for continuous fluid film shapes with simple functions only [References 7, 22-25] and fail to demonstrate the fact that optimum .alpha. values required for the maximum load carrying capacity have different values for infinitely wide and/or narrow bearings. Historically, the first computer method of solving bearing problems was Kingsbury's electrolytic tank method [Reference 26]. A mechanical differential analyzer has also been employed in an attempt to include the actual temperature distribution in the oil film [Reference 27]. The inadequacy of the solutions obtained by these methods has led to finite difference methods.
Various relaxation schemes have been used to accomplish numerical solutions for hydrodynamic thrust bearings by Archibald [Reference 17] and Christopherson [Reference 27] and for the stepped thrust bearings by Kettleborough [Reference 18]. An improved method by using matrix subroutines instead of relaxation schemes for the numerical solution of the general incompressible fluid film lubrication problem was presented by Castelli and Shapiro [Reference 28].
The solution of the Reynolds equation for imcompressible fluids has been obtained by the formulation of coefficient matrices and then using inversion subroutines [References 4, 29]. The use of new methods [Reference 30] to find nodal pressure without inverting any matrix is also known.
In recent years, several papers have been published using finite element methods for solving the Reynolds equation for different types of bearings [References 19-21, 31-34]. Other than complexities involved in the formulations of the two methods, namely finite difference and finite element techniques, two major factors always have to be considered. These are accuracy of the results and computer time involved. The use of variable dx and dz for different nodes by using higher order finite difference forms and new computer aided finite difference design schemes enlarge the scope of possible applications of finite difference method.
Load carrying capacity for centrally pivoted shoe bearings in practice has puzzled persons skilled in the art for many years, and explanations such as variable viscosity, variable density, viscosity changing with pressure and momentum effect at inlet were among the most usually offered theories [Reference 35-38]. Realizing the importance of surface profile, Raimondi and Boyd in their work [Reference 37] assumed an existence of a convex surface profile caused by manufacturing operations, temperature rise and operating load, and emphasized this as the most important factor accounting for the observed load carrying capacity of centrally pivoted shoes.
Raimondi and Boyd [Reference 37] and Abramovitz [Reference 12] both independently explained the working of the centrally pivoted bearing by assuming a convex surface profile and therefore an existence of a converging, diverging film shape in which the inactive diverging portion of the fluid film is utilized in switching the resultant pressure toward the center of the shoe length.